How did the future tense appear in Latin?

The future -b-, and for that matter the imperfect -b-, come from the Indo-European verb for “be”, bhu: The Latin Language. So amabo originally meant “I am to love”, and amabam “I was to love”. The process of words turning into grammatical affixes is called Grammaticalization. And one of the characteristics of grammaticalisation is that it happens over and over, in the same language: French aimerai involves the same process as amabo, but with the verb for “have” rather than “be”.